Column-$I$ represents the formula for ${v_{rms}}$ and Column-$II$ represents the corresponding condition (phenomena). Match them correctly:

Column-$I$	Column-$II$
$(a)$ ${v_{rms}} = \sqrt {\frac{3P}{\rho}}$	$(i)$ For $1 \text{ mole ideal gas}$
$(b)$ ${v_{rms}} = \sqrt {\frac{3RT}{M_0}}$	$(ii)$ For one molecule of gas
$(c)$ ${v_{rms}} = \sqrt {\frac{3{k_B}T}{m}}$	$(iii)$ On the basis of kinetic theory

(A) The root mean square speed $({v_{rms}})$ of gas molecules is derived from the kinetic theory of gases as ${v_{rms}} = \sqrt{\frac{3P}{\rho}}$, where $P$ is pressure and $\rho$ is density. Thus, $(a)$ matches with $(iii)$.
For $1 \text{ mole}$ of an ideal gas, the formula is ${v_{rms}} = \sqrt{\frac{3RT}{M_0}}$, where $R$ is the universal gas constant, $T$ is temperature, and $M_0$ is molar mass. Thus, $(b)$ matches with $(i)$.
For a single molecule of gas, the formula is ${v_{rms}} = \sqrt{\frac{3k_BT}{m}}$, where $k_B$ is the Boltzmann constant and $m$ is the mass of one molecule. Thus, $(c)$ matches with $(ii)$.
Therefore, the correct matching is $(a-iii, b-i, c-ii)$.